A Probability Question.

Factual Questions
1

A bunch of us folks are playing a game over at MPSIMS called Mafia. An aspect of the game concerning probability has been particularly confusing. Since I can’t really trust anyone in that game to tell me the truth I’ve decided to try my luck here.

A Beat Cop is allowed to investigate one person each night in a town. When he investigates anyone he has fifty percent chance of being right. An investigated person can show up as either: Citizen, Mason, Beat Cop, Detective, Guardian Angel, Doctor, Vigilante, Serial Killer or Mafia. When someone is being investigated, the incorrect answers are put through random.org and the result of that is put up against the correct answer in a coin flip.

The town has 22 people. It has 3-6 Mafia members, one Serial Killer, one Guardian Angel, two Masons, one Beat Cop, one other cop that is either a Detective or a Beat Cop and the rest are citizens, that is about 10-13 Citizens. The Doctor and the Vigilante have been killed, and may their souls R.I.P. They can still show up as a result however.

What’s the probability that if an investigated person shows up as a Citizen that he is actually a Citizen? What if he turns up as any of the other roles, does the probability change since there are roughly only one or two players that could hold these roles?

What if the cop investigates someone twice and gets the same result, what are the chances that he is correct?

Thanks in advance.

2

My first thought is that the answer is 1/2. My justification is as follows. No matter how a person shows up, there is a fifty percent chance that the indication is correct, and fifty percent that it is incorrect. So if someone shows up as a Citizen, then there is a fifty percent chance that they are a Citizen, and a fifty percent chance that they are not.

But I’m willing to bet that my first thought here is the wrong thought to have.

-FrL-

3

There are two possible ways of getting a citizen hit. (1) Is that the beat cop found a citizen and he was right, and (2) The beat cop investigated a non-citizen, his investigation was incorrect, and citizen was randomly selected.

The odds of (1) are:

[Select a citizen]*[Investigation is correct]

x/22*.5, where x is the number of citizens.

The odds of (2) are:

[Don’t select a citizen][Investigation Wrong][randomly choose citizen] or

[(22-x)/22].5[1/8]

Then we want to know the percentage of the total citizen hits come from 1, so we do:

[Probability of 1]/[Probability of 1 + Probability of 2] or
[x/22*.5]/[x/22*.5+(22-x)/22*.5*1/8]

For 10 citizens I get 87%, for 11 88.9%, for 12 90.5%, and for 13 92.0% probability that the beat cop has actually found a citizen.

Yes. The probability of a false hit will range from 0 (All players remaining are of the class) to 100% (No players remain of the class). The last equation I gave will provide the % for any class, just change x to the proper number.

The scenarios are the same for this problem, assuming the player remains in the same role, as for the first question. The only difference is that now we are asking what the probability is of them happening two times in a row. For scenario one it’s:

The odds of (1) are:

([Select a citizen]*[Investigation is correct])^2

(x/22*.5)^2, where x is the number of citizens.

The odds of (2) are:

[Don’t select a citizen][Investigation Wrong][randomly choose citizen] or

([(22-x)/22].5[1/8])^2

Then we do:

[Probability of 1]/[Probability of 1 + Probability of 2]

(x/22*.5)^2/((x/22*.5)^2+(((22-x)/22).5(1/8))^2)

For 10 citizens I get 97.7%, for 11 98.5, for 12 98.9, and for 13 99.3%.

4

If the person is a citizen, then there is a 50% chance for them to show up as citizens. If the person is not, they’ve got a 6.25% chance for them to show up as citizens. Currently, about 50% of the folks are citizens, too. Bayes’ theorem says that P(A|B) = P(B|A) * P(A)/P(B), where P(X) is the probability of X, and P(X|Y) is the probability of X if Y is true.

So, the probability that someone is a citizen given that one investigation says they’re a citizen is equal to the probability that an investigation shows that someone is a citizen when the person is a citizen, times the probability that someone is a citizen, divided by the probability that a randomly chosen person would show up as a citizen.

The first value is exactly 50%, the second value is about 50%, and the third value is about 28%. Putting those values into the equation gives a result that there’s about a 90% chance that a person who shows up as a citizen is actually a citizen.

If a person shows up as a citizen twice, then the first value is exactly 25%, the second value is about 50%, and the third value is about 13%, so there’s about a 96% chance that a person who shows up twice as a citizen is actually a citizen.

5

This is sort of like the Monty Hall problem. Let’s say there are 100,000,000 citizens, 1 mason, and 1 doctor remaining. There odds that you pick a citizen are overwhelming, and obviously that’s what you are going to believe no matter what the investigation says. Why? Because the chance that you randomly picked a non-citizen is vanishingly small (1/50,000,000).

6

Waitaminute… doesn’t this take into account information not available to the player?

In other words, isn’t this question being asked from the point of view of a player of the game? So shouldn’t we use only information available to a player of the game?

You and I watching the game, but not playing, may know there are no more Citizens left, for example, and so for us, the probability that “Citizen” is correct is zero. But a player of the game does not know there are no more Citizens left, so to him, the probability is different, right?

-FrL-

7

Got it!

-FrL-

8

Just to clarify: If the investigation is wrong, is the false result returned weighted by the number of those professions? That is to say, if the coin flip fails, is the outcome any more likely to be “citizen” than, say, “doctor”?

9

No weights. Vigiliante(of which is(was) exactly one) is as likely to turn up on a bad investigation as Citizen.

10

I got about 70% when x is 5, but 27% when x is 1. Does that sound correct?

I’m not so confident with my math skills.

11

This a more detailed explanation of the methods being used. It’s also worth pointing out two things that slightly change the results:
A ‘false’ result may still yield the correct answer (1/9 chance), and
the total number of ‘false positives’ is 9, not 8 (I think treis may have left out the beat cop).

Bayes Theorem is
P(A|B) =

P(B|A)*P(A)

P(B|A)*P(A) + P(B|~A)*P(~A)

(’|’ means ‘given’, ‘~’ means ‘not’, P(x) means ‘probability of x’).
Let A = person X is actually role i
Let B = result of investigation of X is role i

Also
r = total number of roles
p = total players
s[sub]i[/sub] = total (unknown) players of role i

The top part of the fraction is
(probability of a correct result)*(prob. that the person is that role)
OR
(1/2 + 1/2r) * (s[sub]i[/sub]/p)

which includes the lucky break ‘wrong’ answer that turns out right.

The left-hand bottom part = the numerator, and the right-hand part is:

(probability of a ‘wrong’ result giving this role)(probability that person is not that role)
OR
(1/21/r)*((p-s[sub]i[/sub])/p)

Which finally results in the formula:

(1/2 + 1/2r) * (s[sub]i[/sub]/p)

(1/2 + 1/2r) * (s[sub]i[/sub]/p) + (1/21/r)((p-s[sub]i[/sub])/p)

You could plug the numbers in directly, but assuming s[sub]i[/sub] > 0, this simplifies to :

(r + 1)

(r + p/s[sub]i[/sub])

(If s[sub]i[/sub] is 0, we know the prob. is 0, and our investigation was fruitless - e.g. someone reads as ‘doctor’ after the doctor is killed.)

Lakai, your numbers are pretty close (try them with these adjustments).

12

I realized the OP didn’t mention that a person could end up correct on a ‘wrong’ result, but that was something NAF1138 (the Mafia moderator) mentioned in that thread.

13

Did he? Where?

I thought NAF1138 only put the wrong answers through random.org and then used that result up against the right answer.

14

Now I have to look it up …

Here it is (p. 4), the third bit says “they could show up as what they really are”:
http://boards.straightdope.com/sdmb/showpost.php?p=8402333&postcount=164

It’s not 100% clear, but that does seem to be what he’s doing. Though in context that sort of implies “Godfather” may come up on a ‘wrong’ answer; maybe it could.

If he changed it since then, I apologize. I’ve only sort of been following along.

Anyway it doesn’t change the answer all that much, given all the other unknowns in that game.

15

Ok, it seems as if he is including the Godfather and The Miller as among the answer choices that can come up. These people if investigated, are supposed to show up as something other than they really are. The Godfather as a Citizen and The Miller as Mafia. That would mean that only the Godfather and the Miller could come up as they are when a beat cop gets the “wrong” result, since the “correct” result is supposed to be wrong.

I suppose that if someone gets the result of Godfather he would have to assume he is wrong or that the Miller blocked the Godfather’s powers and he got the correct result. Though that is way beyond the scope of the question.

I thought it would be best to simplify the question without including these roles. As long as we know that if a person turns up as town the investigation is roughly 90% right, that if a person turns up as mafia the investigation is about 70% correct, and that if some turns up as the Serial Killer the investigation is about 27% correct, then we should have a good understanding of what to do with the results of a Beat Cop investigation.

16

That only applies to the specific case of the GodFather and the Miller, which are sort of special circumstances.